Why does moving left use plus inside the function argument?

Conditions

• The transformation is a horizontal translation of the graph of $y=f(x)$.
• The shift amount is a positive constant.

Reasoning, step by step

1. Recognize that horizontal shifts affect the input variable $x$ inside the function argument.
2. Understand that to move the graph left, the function must receive an input that is $h$ units greater to produce the same output as the original function.
3. Apply the rule: left by $h$ units corresponds to replacing $x$ with $x+h$.
4. Note the speaker's warning that the sign inside the parentheses is opposite the verbal direction.

Example

The speaker explicitly warns that when thinking of moving left from the origin, one naturally thinks of negative 4, “but that’s not what this is,” and then emphasizes that left uses plus inside the function, writing $f(x+4)$.

Common misconceptions

• Assuming that a leftward shift on a number line corresponds to a minus sign inside the function argument.
• Confusing horizontal shifts with vertical shifts, where the sign matches the direction.

Watch the explanation

Explore next

• How do you write the equation for translating $y=f(x) 4$ units left and 3 units down?
• Why does translating right use x-5 instead of $x+5$?